Distributional chaos for weighted translation operators on groups

TL;DR

This paper investigates distributional chaos in weighted translation operators on locally compact groups, providing conditions for chaos, constructing examples, and exploring properties of irregular vectors.

Contribution

It offers a sufficient condition for distributional chaos in weighted translations and analyzes the structure of irregular vectors, advancing understanding in operator dynamics on groups.

Findings

Established a sufficient condition for distributional chaos.

Constructed examples of distributionally chaotic weighted translations.

Analyzed properties of irregular vectors and their subsets.

Abstract

In this paper, we study distributional chaos for weighted translations on locally compact groups. We give a sufficient condition for such operators to be distributionally chaotic and construct an example of distributionally chaotic weighted translations by way of the sufficient condition. In particular, we prove the existence of distributional chaos and Li-Yorke chaos for weighted translations operators with aperiodic elements. Furthermore, we also investigate the set of distributionally irregular vectors ($DIV$) of weighted translations through the cone and equivalence classes. When the field is that of complex numbers, we uncover several properties on certain subsets of $DIV$, including their connectedness and correspondences with some measurable subsets in locally compact groups.